The gradient flow of a generalized Fisher information functional with respect to modified Wasserstein distances

Abstract

This article is concerned with the existence of nonnegative weak solutions to a particular fourth-order partial differential equation: it is a formal gradient flow with respect to a generalized Wasserstein transportation distance with nonlinear mobility. The corresponding free energy functional is referred to as generalized Fisher information functional since it is obtained by autodissipation of another energy functional which generates the heat flow as its gradient flow with respect to the aforementioned distance. Our main results are twofold: For mobility functions satisfying a certain regularity condition, we show the existence of weak solutions by construction with the well-known minimizing movement scheme for gradient flows. Furthermore, we extend these results to a more general class of mobility functions: a weak solution can be obtained by approximation with weak solutions of the problem with regularized mobility.